PHYSICAL REVIEW B 88, 245432 (2013)
Electron-electron interactions in bilayer graphene quantum dots
M. Zarenia,1 B. Partoens,1 T. Chakraborty,1,2 and F. M. Peeters1
1
Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
2
Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2
(Received 1 March 2013; published 20 December 2013)
A parabolic quantum dot (QD) as realized by biasing nanostructured gates on bilayer graphene is investigated
in the presence of electron-electron interaction. The energy spectrum and the phase diagram reveal unexpected
transitions as a function of a magnetic field. For example, in contrast to semiconductor QDs, we find a valley
transition rather than only the usual singlet-triplet transition in the ground state of the interacting system. The
origin of these features can be traced to the valley degree of freedom in bilayer graphene. These transitions have
important consequences for cyclotron resonance experiments.
DOI: 10.1103/PhysRevB.88.245432
PACS number(s): 72.80.Vp, 71.10.Li, 73.21.La, 81.05.ue
I. INTRODUCTION
II. CONTINUUM MODEL
The electronic properties of quantum dots (QDs) in
graphene, a single layer of carbon atoms arranged in a
honeycomb lattice,1–3 have been studied extensively due
to their unique properties and their potential for applications in graphene devices.4–7 Since Klein tunneling prevents
electrostatic confinement in graphene, direct etching of the
graphene sheet is perhaps the only viable option for quantum
confinement. In such systems, controlling the shape and edges
of the dot remains an important challenge but the exact
configuration of the edges is unknown.8 The latter is important
because the energy spectrum depends strongly on the type of
edges.9,10
Two coupled layers of graphene, called bilayer graphene
(BLG), have quite distinct properties from those of a single
layer. In pristine BLG the spectrum is gapless and is approximately parabolic at low energies around the two nonequivalent
points in the Brillouin zone (K and K ). In a perpendicular
electric field, the spectrum displays a gap which can be tuned
by varying the bias.11 Nanostructuring the gate would allow
tuning of the energy gap in BLG, which can be used to
electrostatically confine QDs12,13 and quantum rings.14 Here
the electrons are displaced from the edge of the sample and,
consequently, edge disorder and the specific types of edges
are no longer a problem. Such gate-defined QDs in BLG were
recently fabricated by different groups,15–17 who demonstrated
experimentally the confinement of electrons and Coulomb
blockade.
In the present work we investigate the energy levels of a
parabolic QD in BLG in the presence of Coulomb interaction.
Here we consider the two-electron problem as the most simple
case to investigate the effect of electron-electron correlations.
Similar studies have been reported for semiconducting QDs
over the last two decades18,19 and recently for graphene
QDs20 and graphene rings.21 At present, no similar study
has been reported for BLG quantum dots. An important
issue for graphene structures is the extra valley-index degree
of freedom where the electrons have the possibility to be
in the same valley or in different valleys.22,23 Here we
show that the competition between the valley index and
the electron spin leads to unique behaviors that shed light
on the fundamental properties of the ground-state energy of
BLG QDs.
In order to find the single-particle energy spectrum of a
parabolic QD we employ a four-band continuum model to
describe the BLG sheet. In the valley-isotropic form,24 the
Hamiltonian is given by
⎛
⎞
τ U (r)
π
t
0
⎜ π†
τ U (r)
0
0 ⎟
⎜
⎟
(1)
H =⎜
⎟,
⎝ t
0
−τ U (r)
π† ⎠
1098-0121/2013/88(24)/245432(5)
0
0
π
−τ U (r)
where t ≈ 400 meV is the interlayer coupling term. The
additional coupling terms which lead to the trigonal warping
effect are neglected. The trigonal warping effect is only
relevant at very low energies (i.e., E < 2 meV) in the absence
of an electrostatic potential.24 π = −ivF eiθ [∂r + i∂θ /r −
(eB/2)r] is the momentum operator in polar coordinates and
in the presence of an external magnetic field B, and vF =
106 m/s is the Fermi velocity. The valley-index parameter
τ distinguishes the energy levels corresponding to the K
(τ = +1) and the K (τ = −1) valleys. The electrostatic
potential U (r) is applied to the upper layer and −U (r) to
the lower layer. For the QD profile we consider a parabolic
potential U (r) = Ub (r/R)2 , where the potential Ub and the
radius R define the size of the dot (Fig. 1). The eigenstates of
the Hamiltonian (1) are the four-component spinors,
ψ(r,θ ) = eimθ [φA (r),e−iθ φB (r),φB (r),eiθ φA (r)]T ,
(2)
where φA,B,B ,A are the envelope functions associated with
the probability amplitudes at the respective sublattice sites of
the upper and lower graphene sheets and m is the angular
momentum. The orbital angular momentum Lz does not
commute with the Hamiltonian and is no longer quantized.
This is different from two-dimensional semiconductor QDs,
where [H,Lz ] = 0. However, the wave function ψ(r,φ) is an
eigenstate of the operator
0
σz
−I 0
,
(3)
+
Jz = Lz +
2 0 I
2 0 −σz
with eigenvalue m, where I is the 2 × 2 unitary matrix and
σz is one of the Pauli matrices. The first operator inside the
bracket is a layer index operator, which is associated with the
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©2013 American Physical Society
ZARENIA, PARTOENS, CHAKRABORTY, AND PEETERS
PHYSICAL REVIEW B 88, 245432 (2013)
A. Single-particle energy levels
Figure 2(a) shows the lowest single-electron energy levels
as a function of the magnetic field for a QD with Ub =
150 meV and R = 50 nm. The energy levels are labeled
by their angular momentum and their valley index (m,τ ).
We begin with the B = 0 case. Notice that the singleparticle ground state does not have m = 0 as expected for
semiconductor QDs, but instead has the momentum m = 1 at
K and m = −1 at K in agreement with Ref. 12.
For large magnetic fields the eigenstates are strongly localized at the origin of the dot, where U → 0. Therefore, the
spectrum approaches the Landau levels (LLs) of an unbiased
BLG [black dotted curves in Fig. 2(a)] and consequently
some of the energy levels approach E = 0 as the field
increases. Notice that this is quite distinct from semiconductor
QDs, where the zeroth LL is absent and thus the energy
of the confined stats, i.e., the Fock-Darwin states, increase
with magnetic field. Breaking of the electron-hole symmetry
due to the presence of both electric and magnetic fields
lifts the valley-degeneracy in nonzero magnetic fields. The
energy spectrum also displays the EK (m,B) = EK (−m, −B)
symmetry, which is another feature that is unique to BLG
QDs. This symmetry is a consequence of the fact that the QD
is produced by a gate that introduces an electric field and thus a
preferential direction. Inserting (m,β) → (−m, −β) and τ →
−τ in Eqs. (4) and using Eτ (m,B) = E−τ (−m, −B), one can
find the relations φAK (m,B) = φBK (−m, −B) and φBK (m,B) =
φAK (−m, −B) between the wave function components of the
K and K valleys.
B
r
Bilayer graphene
FIG. 1. (Color online) Schematic illustration of the potential
profile for a parabolic bilayer graphene quantum dot.
behavior of the system under inversion, whereas the second
one denotes the pseudospin components in each layer.
Solving the Shrödinger equation, H = E, the radial
dependence of the spinor components is described by
d
m
+ + βr φA
dr
r
(m − 1)
d
−
− βr φB
dr
r
d
(m + 1)
+
+ βr φA
dr
r
m
d
− − βr φB dr
r
= −[E − τ U (r)]φB ,
= [E − τ U (r)]φA − tφB ,
= [E + τ U (r)]φB − tφA ,
= −[E + τ U (r)]φA ,
(4)
where β = (eB/2)R 2 is a dimensionless parameter. The
energy, the potential, and the hopping term t are written in
units of E0 = vF /R, with R being the unit of length. The
coupled equations (4) are solved numerically using the finite
element method.25
80
B. Two-electron energy spectrum
The Hamiltonian describing the two-electron system
is given by HT = H (r1 ) + H (r2 ) + Vc (r1 − r2 ), where
(b) two non-interacting electrons
(a) single-electron
30
70
(2,
,0
(-3
(0,0)
50
)
20
(-2
,0
40
)
30
15
3
1
-10
0
2
-4
-2
-2
2
-1
1
-5
,-2
)
)
0
5
)
10 3
0
-15
0
4
5
(-3
,-2
,2
20
(-2
(-2
Energy (meV)
2)
25
60
-3
-5
-3
10
-1
10
15
B (Tesla)
0
2
4
6
8
10
B (Tesla)
FIG. 2. (Color online) Energy spectrum of a parabolic QD in BLG with R = 50 nm and Ub = 150 meV as a function of the magnetic field
for (a) a single particle and (b) two noninteracting electrons. The lowest energy levels in (a) are labeled by the angular momentum m. The
levels corresponding to the K and K valleys are, respectively, shown by the blue solid and red dashed curves. The black dotted curves are the
LLs of a BLG sheet. The levels in (b) are labeled by (M,T ), where M = m1 + m2 is the total angular momentum and T = τ1 + τ2 is the total
valley index. Levels having the same valley index are plotted using the same type of curve.
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ELECTRON-ELECTRON INTERACTIONS IN BILAYER . . .
PHYSICAL REVIEW B 88, 245432 (2013)
Vc = e2 /(4π κ|r1 − r2 |) is the Coulomb interaction between
the two electrons, with κ being the dielectric constant of BLG.
In our calculations we set κ = 3.9, which is the dielectric
constant of gated BLG on top of a hexagonal boron nitride (hBN) substrate.26 We carry out an exact diagonalization of the
above Hamiltonian to obtain the eigenvalues and eigenstates
of the two-electron system. The corresponding two-electron
wave function with fixed total angular momentum M and total
valley index T is constructed as linear combinations of the
one-electron wave functions,
(r1 ,r2 ) =
Ns
Ns i
Cij i (r1 ) ⊗ j (r2 ),
(5)
j
where is an eight-component wave function which is
K = [ψK ,0,0,0,0]T , corresponding to the K valley, and
K = [0,0,0,0,ψK ]T , corresponding to the K valley.27 The
four-component wave function ψK(K ) is given by Eq. (2).
Notice that the two-electron wave function (r1 ,r2 ) has
64 components. The subscripts i ≡ (mi ,τi ) and j ≡ (mj ,τj )
correspond to the one-electron energy levels where the
summations in Eq. (5) are such that the relations M = mi + mj
and T = τi + τj are satisfied. In our calculations Ns , i.e., the
number of lowest single-particle states, is chosen sufficiently
large to guarantee the convergence of the energies. The
singularity due to the 1/|r1 − r2 | term in the matrix elements
is avoided by using an alternative expression in terms of the
Legendre function of the second kind of half-integer degree.28
In Fig. 3, we show two representative spectra for two
interacting electrons in a BLG QD with radius (a) R =
20 nm, and (b) R = 50 nm, and Ub = 150 meV. To clearly
see the effect of electron correlations, the spectra for two
noninteracting electrons in a dot with R = 50 nm is shown in
Fig. 2(b) for comparison. The levels are labeled by (M,T ,S)
with M = m1 + m2 the total angular momentum, T = τ1 + τ2
the total valley index, and S the total spin. Energy levels
with the same T are drawn using the same type of curve.
Two electrons can form a nondegenerate singlet state (S = 0)
and a threefold-degenerate triplet state (S = 1). In case the
quantum number S is omitted, the singlet and triplet states
are degenerate. In the following discussion it is useful to
characterize the many-body state by the single-particle basis
function in expression (5), which has the largest contribution.
We denote the basis function in which the first and second
electrons have, respectively, angular momentum m1 and m2
and valley index τ1 and τ2 as (m1 ,τ1 ) ⊗ (m2 ,τ2 ) ≡ φm1 ,τ1 ⊗
φm2 ,τ2 .
The spectra of the two interacting electrons in Figs. 3(a)
and 3(b) are a result of three competing effects. It is evident
that the energy of the single-particle states as a function of
the magnetic field, shown in Fig. 2(b), determines partly
the spectrum. However, turning on the Coulomb interaction
between the electrons changes the spectrum drastically. While
the noninteracting state (−2, −2) ≡ (−1, −1) ⊗ (−1, −1) is
the ground state, the many-body interacting state in which this
single-particle basis function has the largest contribution is
an excited state and does not even appear in Figs. 3(a) and
3(b). Instead the many-body state (0,2,1) is the ground state
for small magnetic field values, with the main contribution
(1,1) ⊗ (−1,1). The Coulomb interaction is clearly not a small
perturbation. In Fig. 3(c), the evolution of the average distance
between both electrons is shown for different single-particle
basis states (for the R = 50 nm case). This difference in the
average distance can be understood from the single-particle
densities. While single-particle states (−1, −1) and (1,1)
have a nonzero density in the origin, the density of the
single-particle state (−1,1) is zero in the origin. Therefore,
this average distance is much larger, and consequently the
Coulomb interaction is much lower, for the basis function
(1,1) ⊗ (−1,1) than for (−1, −1) ⊗ (−1, −1), which is the
reason why the many-body state (0,2,1) has a lower energy.
A more subtle effect is played by the exchange interaction.
As mentioned, the ground state at small magnetic field values
is given by the many-body state (0,2,1), which is a triplet
state. The corresponding singlet state (0,2,0) is slightly higher
in energy. Also note that the state (−2,0), with the main singleparticle contribution (−1,1) ⊗ (−1, −1) is higher in energy at
small magnetic fields, although the Coulomb interaction contribution is expected to be very similar [compare curves (0,2,1)
and (−2,0) in Fig. 3(c)]. The reason is again the exchange
FIG. 3. (Color online) (a),(b) The same as Fig. 2(b) but in the presence of Coulomb interaction and for (a) R = 20 nm and (b) R = 50 nm.
The levels are labeled by (M,T ,S) where S indicates the total spin. The curves with the same total valley index T are shown with the same
type of curve. (c) The average distance between the electrons as function of magnetic field for different single-particle basis functions. (d) The
radius-magnetic field (R − B) phase diagram of the ground-state energy of a two-electron BLG QD.
245432-3
ZARENIA, PARTOENS, CHAKRABORTY, AND PEETERS
PHYSICAL REVIEW B 88, 245432 (2013)
FIG. 4. (Color online) The radial electron density for the two-electron QD of Fig. 3(b) for (a) (M,T ) ≡ (−2,0) and (b) (M,T ,S) ≡ (−3,2,1)
and for B = 0, 1.5, 5 T. The upper insets show the total pair-correlation function for B = 0 T and B = 5 T. The lower insets show separate
parts of the pair-correlation function for B = 0. The black dot indicates the position of the first electron, which is pinned at r1 = (0.4R,0).
interaction energy gain for the triplet state (0,2,1). State (−2,0)
is fourfold degenerate: The triplet configuration does not lead
to an exchange energy gain because both electrons occupy
different valleys. Only for larger magnetic field values, the state
(−2,0) takes over from the state (0,2,1) to become the ground
state. This is caused by the evolution of the single-particle
energies. The next state that becomes the ground state with
increasing magnetic field is the singlet (−2,2,0), with the
main single-particle contribution (−1,1) ⊗ (−1,1). Because
twice the same single-particle level is occupied, no exchange
energy gain is possible. Nevertheless, this state becomes
the ground state due to the evolution of the single-particle
energies, together with the fact that the average distance
between both electrons is even smaller [see Fig. 3(c)], as both
electrons occupy a single-particle level with zero density in
the origin. With increasing field, it becomes beneficial for
an electron to jump from the single-particle level (−1,1) to
the single-particle level (−2,1), resulting in the triplet state
(−3,2,1) with the main contribution (−2,1) ⊗ (−1,1).
While in conventional semiconductor QDs, the ground state
shows a series of singlet-to-triplet transitions as a function of
the magnetic field strength,18,19 a more complex phase diagram
is found for BLG QDs. In Fig. 3(d) we plot this phase diagram
for the same Ub and dielectric constant as used for Figs. 3(a)
and 3(b). For small magnetic field values, the ground state
is found to be a triplet state. With increasing field a valley
transition occurs, resulting in a fourfold-degenerate state. Next,
again a valley transition occurs into a singlet state. Further
increasing the magnetic field favors again a triplet state.
The electron density, ρ(r) = 2i=1 δ(r − ri ) is shown in
Figs. 4(a) and 4(b), respectively, for the (−2,0) and (−3,2,1)
states of a two-electron BLG QD with R = 50 nm and
Ub = 150 meV and for three values of the external magnetic
field B = 0,1.5,5 T. Comparing the density profiles in (a) and
(b), the maximum of the density is shifted towards higher
radial distance in the state (−3,2,1). This is a consequence of
the fact that the electrons in the many-particle (−3,2,1) state
occupy single-particle states with higher angular momentum.
As the magnetic field increases, the electrons are pulled
closer towards the center of the dot. The upper insets in
Figs. 4(a) and 4(b) show the total pair-correlation function
P = |(r,r2 )|2 = P1,1 + P1,2 for B = 0 T and B = 5 T. The
P1,1 and P1,2 terms refer, respectively, to the contribution of
the sublattices in the same layer and in the different layers. In
the lower insets the terms P1,1 and P1,2 are plotted separately
for B = 0 T, which, respectively, indicate the probability to
find the second electron in the same layer or in the other layer
if the first electron is fixed at a certain point.
In cyclotron resonance experiments, transitions are induced between the ground state and excited states. For
the BLG QD the selection rule on the change of total
momentum m = ±1 is still fulfilled. This is apparent when
we calculate the transition energies and the corresponding
transition rates for dipole transitions using the relation
FIG. 5. (Color online) The cyclotron transition energies for the
QD of Fig. 3(b). The transition energies are labeled by their final
states: (M,T ,S) for the triplet (or singlet) states and (M,T ) for the
degenerate single-triplet states. Levels having the same total valley
index are plotted using the same type of curve.
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ELECTRON-ELECTRON INTERACTIONS IN BILAYER . . .
PHYSICAL REVIEW B 88, 245432 (2013)
|i | 2j =1 rj exp (±iθj )/2|f |2 . This relation also implies
conservation of the total spin, i.e., S = 0. The valley degree
of freedom dictates new transition rules for BLG QDs, i.e.,
T = 0 or T = ±2. This means that those transitions are
possible in which at least one electron remains in the same
valley during the transition. The lowest possible transition
energies for a two-electron QD with R = 50 nm and Ub =
150 meV are shown in Fig. 5. The possible transitions
are labeled by the final states (M,T ,S). The discontinuities
between the transition energies at B ≈ 0.5 T, B ≈ 0.5 T, and
B ≈ 7 T are due to the valley and singlet-triplet transitions
[see Fig. 3(b)].
III. CONCLUDING REMARKS
In summary, we have investigated the energy levels,
the electron density, the pair correlation function, and the
cyclotron transition energies of electrostatically confined QDs
containing one or two electrons in a BLG. Such QDs can be
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ACKNOWLEDGMENTS
This work was supported by the Flemish Science Foundation (FWO-Vl), the European Science Foundation (ESF)
under the EUROCORES program EuroGRAPHENE (project
CONGRAN), and the Methusalem foundation of the Flemish
Government. T.C. is supported by the Canada Research Chairs
program of the Government of Canada.
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The total single-particle wave functions can be expressed as a
superposition of the contribution of the two valleys via the Bloch
wave function, i.e., exp(iK · r)K and exp(iK · r)K . When both
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and when the electrons belong to different valleys the corresponding
K
matrix element becomes zero K (r1 ) ⊗ K
j (r2 )|Vc | (r1 ) ⊗
K
(r2 ) = 0. Thus, the Bloch wave functions are not important
when constructing the matrix elements.
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